Three Boys

Alan: Hi, Bill. I hear you just adopted three boys. How old are they?
Bill: If you multiply their ages together you get 36.
Alan: I can't work it out from that.
Bill: Well, if you add their ages together, you get the number of the house you used to live in when you lived in Salisbury.
Alan: Hmm, I still can't work it out.
Bill: Well, the eldest boy plays the harpsichord.
Alan: Ah! Now I know.

How did Alan work out the ages of the three boys, and what are their ages?

Hint

Look at all the combinations of ages that give 36 when multiplied together - there's not as many as you might think.
Even though WE don't know the number of Bill's old house, presumably he and Alan do.Hide

Answer

Their ages are 9, 2 and 2.
There are eight possible combinations that have a product of 36. Each combination has a unique sum, except for two; 6,6 and 1, and 9, 2 and 2. They both add up to 13. Since Alan still couldn't solve the problem even after being effectively told what the sum was, we must assume the number of the house was indeed 13.
Being told the ELDEST plays the harpsichord tells us there IS an eldest, so the combination must be 9, 2 and 2.Hide

Comments

If the three boys had been Bill's natural children, we could be sure that at 1, 6 and 6 there would still be an eldest--even if only by a few minutes.
However, don't forget we are told they are adopted and hence it is possible that two 6-year-olds could be IDENTICAL in age. Only with 2, 2 and 9 can we be sure there is an eldest.